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Ordinal notations (part 2) – θ function

θ function is a binary function. It’s defined as follows:

Where and means the αth uncountable cardinal. is the first uncountable ordinal and has cardinality , is the smallest ordinal that has cardinality . Generally is the smallest ordinal that has cardinality . They’re so large that , and for all .

It means that, θ(α,β) is the (1+β)-th ordinal such that it cannot be built from ordinals less than it by addition, applying θ(δ,_) where δ < α and getting an uncountable cardinal.

Explanation

To get we need to deal with C(0,α). C0(0,0) already contains 0, so let’s begin from 1. For C(0,1), C0(0,1) only contains 0, adding and getting uncountable cardinals cannot get 1, so . For C(0,2), 1 is in it, so 2=1+1 is in it. And so on, 3 is in C(0,3), 4 is in C(0,4) ,etc. Until ω, it cannot be built from natural numbers by addition and getting uncountable cardinals, so . Further, .

is the least ordinal that can’t be built from ordinals less than it by addition and applying (and getting uncountable cardinals) – that’s where the Cantor’s normal form cannot get. So . Further, .

It seems that below , making θ function an extension of φ function. Even is true.

Beyond Feferman–Schütte ordinal

However, things suddenly go wrong. In , we cannot use , we can just use for , so .
From on, we can use in , so . And further, .

Then for , and for – It meets another problem at .

, and . Next, , and . Now this pattern where will continue up to Ω, so it’s enough to define larger ordinals. But we cannot name, say, what’s the first fixed point of , what’s LVO, what’s BHO, etc.

To express large ordinals only using θ function (and Cantor’s normal form), we need to deal with something beyond . Using the result above, for all α < Ω. Now notice that C(Ω+1,α) contains Ω and can apply θ(Ω,α) in it, so is in . In fact, is the (1+β)-th fixed point of .

It seems that, if we write θ function in this way: , where with and , then it’s the (1+β)-th fixed point of where and itself. This pattern holds up to – even beyond Bachmann-Howard ordinal. For example, is the first fixed point of , which is called large Veblen ordinal. The small Veblen ordinal is , and the Bachmann-Howard ordinal is .

Beyond Bachmann-Howard ordinal

The (1+β)-th fixed point of is . For , but – The same thing at happens at Ω now.

Another important point is . What is it? cannot be built from ordinal less than it and Ω by addition and where , so it’s . What’s more, for .
Then is the 2nd ordinal that cannot be built from ordinal less than it and Ω by addition and where – it’s . What’s more, for β < Ω and ., and . And for . Note that for any α!

The next step is . Since , it can be built from ordinals below it and Ω and Ω2 by addition and where , so it’s not – there’re many things between them, such as , , , , etc.

Then we turn to . For β < Ω2 and , but there’re many things between and .

Beyond θ(Ωω,0)

The “getting an uncountable cardinal” in definition start to works now. is the supremum of all the for natural number n.

is not because we have , then , then in . In fact, is the (1+β)-th fixed point of .